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Unit 2: Normal DistributionsPowerPoint Presentation

Unit 2: Normal Distributions

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### Unit 2: Normal Distributions

Alexis, Sommy, Elizabeth, Charlie

Density Curves

- Always on or above the horizontal axis
- Has an area of exactly one underneath it
- Describes the overall pattern of the distribution and shows the proportion of all observations within a certain interval
- Median: “equal areas point”, divides the area of the curve in half
- Mean: “balance point”, the point at which the curve would balance if made of sold material

Solving problems with density curves

- Plot your data, make a graph, usually a histogram or stem plot
- Look for a pattern (SOCS)
- Calculate a numerical summary to describe center and spread

Empirical Rule

- In a normal distribution with mean µ and standard deviation σ
- 68% of the observations fall within σ of the mean µ
- 95% of the observations fall within 2σ of the mean µ
- 99.7% of the observations fall within 3σ of the mean µ

Percentile

- Pth percentile is the value such that p percent of the observations fall at or below it
- Often used for test scores in which the data is normally distributed

Comparing Distributions

- Z-scores
- Percentiles: p percent of values that fall at or below the given number
- Can use either z-scores or percentiles to compare data across two different distributions

Standardizing Data

- Z score: tells how many standard deviations away from the mean and the original observation falls and in which direction
- Observations larger than the mean are positive when standardized
- Observations smaller than the mean are negative when standardized

Standard Normal Table

- Gives the area under the standard normal curve
- The table entry value z is the area under the curve left of z

Normal Proportions Calculations

- State the problem in terms of the observed variable x
- Draw a picture of the distribution and shade the area of interest under the curve
- Standardize x to restate the problem in terms of the normal variable z. Draw a picture to show the area of interest under the standard normal curve
- Use the table to find the required area under the standard Normal curve
- Write your conclusion in context

Normal Probability Plot

- Used to see if a normal model is adequate for the data
- If the points lie close to a straight line the plot indicates that the data are Normal
- Systematic deviations from a straight line indicate a Non-Normal distribution
- Outliers appear as points that are far from the overall pattern of the plot

Calculator Keystrokes

- Histogram
- STAT > 1. Edit > ENTER > L1>ENTER> (enter data) > 2nd Y = > 1. Plot > ENTER > On> ENTER> arrow over to histogram drawing> ENTER> arrow down to X list> L1 (second 1)> GRAPH
- Z-score
- 2nd VARS> 2. Normalcdf( > lower (lowest score) >ENTER> upper( maximum score) > ENTER> mean of data> ENTER > standard deviation of data > ENTER> paste> ENTER> answer is z score from table
- Score needed
- 2nd VARS> 3. invNorm( > area (z-score) >ENTER> mean of data> ENTER > standard deviation of data > ENTER> paste> ENTER> answer is score needed to lie in that area of distribution
- Five Number Summary
- STAT > 1. Edit > ENTER > L1>ENTER> (enter values) > STAT> arrow right to CALC> 1. 1-VAR Stats> ENTER> List (L1)> arrow down to calculate> ENTER
- (mean, standard deviation, min, Q1, Median, Q3, Max)

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